The shoe sizes of five students are: 10, 9, 11 and 4. Find the following: a) The median shoe size b) The modal shoe size c) Calculate the mean shoe size.

Shoe sizes: $4, 9, 9, 10, 11$

(a) Median shoe size

Step 1: Arrange the data in ascending order.

Ordered list: $4, 9, 9, 10, 11$

Step 2: Identify the median position.

Number of values $n=5$ (odd), so median is the $\frac{n+1}{2}$th value:

$\frac{5+1}{2} = 3$

Step 3: Select the 3rd value.

3rd value $=9$

Median shoe size: $\boxed{9}$

(b) Modal shoe size

Step 1: Find the frequency of each shoe size.

$4$ appears $1$ time,
$9$ appears $2$ times,
$10$ appears $1$ time,
$11$ appears $1$ time.

Step 2: Select the value with highest frequency.

Highest frequency $=2$ (shoe size $9$)

Modal shoe size: $\boxed{9}$

(c) Mean shoe size

Step 1: Mean formula

$\bar{x} = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5}$

Step 2: Substitute values.

$\bar{x} = \frac{4 + 9 + 9 + 10 + 11}{5}$

Step 3: Calculate sum, step-by-step.

$4 + 9 = 13$

$13 + 9 = 22$

$22 + 10 = 32$

$32 + 11 = 43$

Step 4: Divide by $n=5$.

$\bar{x} = \frac{43}{5} = 8.6$

Mean shoe size: $\boxed{8.6}$